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Theorem fusgrvtxfi 27087
Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
Hypothesis
Ref Expression
isfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
fusgrvtxfi (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi
StepHypRef Expression
1 isfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 27086 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
32simprbi 500 1 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cfv 6328  Fincfn 8484  Vtxcvtx 26767  USGraphcusgr 26920  FinUSGraphcfusgr 27084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-fusgr 27085
This theorem is referenced by:  fusgrfupgrfs  27099  nbfusgrlevtxm1  27145  nbfusgrlevtxm2  27146  nbusgrvtxm1  27147  uvtxnm1nbgr  27172  cusgrm1rusgr  27350  wlksnfi  27671  fusgrhashclwwlkn  27842  clwwlkndivn  27843  fusgreghash2wsp  28101  numclwwlk3lem2  28147  numclwwlk4  28149
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